Download MATH 105 Exam 1: Calculus Problem Solving and more Exams Calculus in PDF only on Docsity!
MATH 105 EXAM 1 October 7, 2005
Name:
Your grade is based on the process as well as the final result. Show all your steps clearly so you will be eligible for the most partial credit. You may use a calculator, but no notes, books, or other students. Good luck!
1.) (15 pts.)
a.) (5 pts.) Sketch the piecewise function
f(x) =
x√ + 2 if x < − 2 x^2 − 4 if − 2 ≤ x ≤ 2 ln(x − 1) if x > 2
b.) (2 pts. each) For the function in part (a.), find the following limits:
i.) lim x→− 2 −^
f(x)
ii.) lim x→− 2 +
f(x)
iii.) lim x→− 2
f(x)
iv.) lim x→ 0 f(x)
v.) lim x→ 2 f(x)
2.) (15 pts.) Compute the derivative function y′^ of y = 3x^2 + 4 using the limit definition of the derivative.
4.) (15 pts.) A tank initially contains 100 gallons of water and 10 pounds of salt, thoroughly mixed. Pure water is added at the rate of 5 gallons per minute and the mixture is drained off at the same rate. (Assume complete and instantaneous mixing.) Thus, S(t), the amount of salt in the tank at time t, is the solution of the IVP S′^ = − 0. 05 S, S(0) = 10.
a.) (5 pts.) Explain why this IVP describes the given process.
b.) (5 pts.) Use calculus to show that S(t) = 10e−^0.^05 t^ is the solution of this IVP. Remem- ber, you have to show both that the solution matches the DE, and that the solution matches the initial condition.
c.) (5 pts.) How much salt is left in the tank after 1 hour?
5.) (15 pts.) Given the function f(x) = sin x, find f′(π) in two ways:
a.) (10 pts.) Zoom in numerically. Show your table of values, and state the conclusion it leads you to.
b.) (5 pts.) Zoom in graphically. Sketch the zoomed graph that indicates the appropriate derivative value. Include the scale (x and y values) on your sketch.
7.) (10 pts.) Sketch the graph of f(x) = x^2 /^3. (It’s OK to use your calculator to help.) Use your graph to answer the following questions.
a.) (2 pts.) For which x-value(s) is f(x) NOT differentiable?
b.) (2 pts.) For which x-value(s) is f(x) increasing?
c.) (2 pts.) For which x-value(s) is f′(x) increasing?
d.) (2 pts.) For which x-value(s) is f(x) concave down?
e.) (2 pts.) For which x-value(s) is F (x), the antiderivative of f(x), increasing?
